An angular momentum conserving Affine-Particle-In-Cell method

TL;DR

The paper introduces APIC, a novel Particle-In-Cell method that conserves both linear and angular momentum by representing particle velocities as locally affine, reducing numerical diffusion and avoiding FLIP-like instabilities.

Contribution

APIC is a new PIC technique that achieves momentum conservation with lumped mass and retains PIC filtering properties, improving stability and accuracy.

Findings

APIC conserves linear and angular momentum during transfers.

APIC reduces numerical diffusion compared to traditional PIC.

APIC avoids velocity instabilities seen in FLIP.

Abstract

We present a new technique for transferring momentum and velocity between particles and grid with Particle-In-Cell (PIC) calculations which we call Affine-Particle-In-Cell (APIC). APIC represents particle velocities as locally affine, rather than locally constant as in traditional PIC. We show that this representation allows APIC to conserve linear and angular momentum across transfers while also dramatically reducing numerical diffusion usually associated with PIC. Notably, conservation is achieved with lumped mass, as opposed to the more commonly used Fluid Implicit Particle (FLIP) transfers which require a 'full' mass matrix for exact conservation. Furthermore, unlike FLIP, APIC retains a filtering property of the original PIC and thus does not accumulate velocity modes on particles as FLIP does. In particular, we demonstrate that APIC does not experience velocity instabilities that are characteristic of FLIP in a number of Material Point Method (MPM) hyperelasticity calculations. Lastly, we demonstrate that when combined with the midpoint rule for implicit update of grid momentum that linear and angular momentum are exactly conserved.